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Lenny Jones

Publications and source records attributed to Lenny Jones.

At least 19 recordsLinked to original sources
arXiv2026

Generalized Wieferich primes and monogenic trinomials

Let $b\ge 2$ be an integer and let $p\ge 3$ be a prime. We say that $p$ is a {\em generalized Wieferich prime base $b$}, or more succinctly, a {\em base-$b$ Wieferich prime,} if $b^{p-1}\equiv 1 \pmod{p^2}$. When $b=2$, $p$ is also known simply as a Wieferich prime. Let $f(x)\in {\mathbb Z}[x]$ be a monic polynomial of degree $N\ge 2$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. Recently, the third author proved that $x^{2p}+2x^p+2$ is monogenic if and only if $p$ is not a Wieferich prime. In this article, we generalize this result to $x^{2n}+bx^n+b$ with certain restrictions on $b\ge 2$ and $n\ge 3$.

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arXiv2026

A Note on Abelian Monogenic Trinomials

An abelian monogenic polynomial $f(x)\in {\mathbb Z}[x]$ is a monic polynomial of degree $N$ that is irreducible over ${\mathbb Q}$, such that the Galois group of $f(x)$ over ${\mathbb Q}$ is abelian, and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we determine all abelian monogenic trinomials of the form $x^{2n}+ax^{n}+b$, where $n,a,b\in {\mathbb Z}$ with $n\ge 1$ and $ab\ne 0$.

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arXiv2026

Wieferich Primes and Monogenic Trinomials

A prime $p$ is called a Wieferich prime if $2^{p-1}\equiv 1 \pmod{p^2}$. A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we show that ${\mathcal F}_p(x):=x^{2p}+2x^{p}+2$ is monogenic if and only if $p$ is not a Wieferich prime.

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arXiv2026

Characterizing monogenic trinomials $\boldsymbol{x^{12}+ax^6+b}$ according to their Galois groups

Let $f(x)=x^{12}+ax^{6}+b\in {\mathbb Z}[x]$, with $ab\ne 0$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots,\theta^{11}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. These results extend recent work on monogenic power-compositional quartic and sextic trinomials.

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arXiv2026

On the monogenicity and Galois groups of $\boldsymbol{x^{2p}+ax^p+b^p}$

Let $f(x)=x^{2p}+ax^p+b^p$, where $p$ is a prime and $a,b\in {\mathbb Z}$ with $ab\ne 0$. If $f(x)$ is irreducible over ${\mathbb Q}$, we say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{2p-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we give a characterization of the monogenic trinomials $f(x)$ according to their Galois groups. These results extend prior investigations of the authors.

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arXiv2026

Monogenic even sextic trinomials and their Galois groups

Let $f(x)=x^6+Ax^{2k}+B\in {\mathbb Z}[x]$, with $A\ne 0$ and $k\in \{1,2\}$. We say that $f(x)$ is {\em monogenic} if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each value of $k$ and each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also determine when these descriptions provide infinitely many such trinomials, and we investigate when these trinomials generate distinct sextic fields. These results extend recent work on monogenic power-compositional sextic trinomials of the form $g(x^3)$ to the situation $g(x^2)$, and thereby complete the characterization, in terms of their Galois groups, of monogenic power-compositional sextic trinomials.

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arXiv2025

Monogenic Strictly-Perron Polynomials

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. A strictly-Perron polynomial is the minimal polynomial of a Perron number $\lambda$ such that $\lambda$ is neither a Pisot number, an anti-Pisot number, nor a Salem number. For any natural number $n\ge 2$, we prove that there exist infinitely many monogenic strictly-Perron polynomials of degree $n$.

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arXiv2025

Monogenic sextic trinomials $x^6+Ax^3+B$ and their Galois groups

Let $f(x)=x^6+Ax^3+B\in {\mathbb Z}[x]$, with $A\ne 0$, and suppose that $f(x)$ is irreducible over ${\mathbb Q}$. We define $f(x)$ to be {\em monogenic} if $\{1,\theta,\theta^2,\theta^3,\theta^4,\theta^{5}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ of $f(x)$ over ${\mathbb Q}$, we use a theorem of Jakhar, Khanduja and Sangwan to give explicit descriptions of all monogenic trinomials $f(x)$ having Galois group $G$. We also investigate when these trinomials generate distinct sextic fields.

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arXiv2025

Monogenic Cyclic Polynomials in Recurrence Sequences

Let $f(x)\in {\mathbb Z}[x]$ be an $N$th degree polynomial that is monic and irreducible over ${\mathbb Q}$. We say that $f(x)$ is {\em monogenic} if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. We say that $f(x)$ is {\em cyclic} if the Galois group of $f(x)$ over ${\mathbb Q}$ is the cyclic group of order $N$. In this article, we investigate the appearance of monogenic cyclic polynomials in certain polynomial recurrence sequences.

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arXiv2025

Monogenic Reciprocal Quartic Polynomials And Their Galois Groups

Suppose that $f(x)=x^4+Ax^3+Bx^2+Ax+1\in {\mathbb Z}[x]$. We say that $f(x)$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\theta^3\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. For each possible Galois group $G$ that can occur in the two cases of $A\ne 0$ with $B=0$, and $AB\ne 0$, we determine all monogenic polynomials $f(x)$ with Galois group $G$.

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arXiv2025

Monogenic Even Cyclic Sextic Polynomials

Suppose that $f(x)\in {\mathbb Z}[x]$ is monic and irreducible over ${\mathbb Q}$ of degree $N$. We say that $f(x)$ is monogenic if $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$, and we say $f(x)$ is cyclic if the Galois group of $f(x)$ over ${\mathbb Q}$ is isomorphic to the cyclic group of order $N$. In this note, we prove that there do not exist any monogenic even cyclic sextic binomials or trinomials. Although the complete story on monogenic even cyclic sextic quadrinomials remains somewhat of a mystery, we nevertheless determine that the union of three particular infinite sets of cyclic sextic quadrinomials contains exactly four quadrinomials that are monogenic with distinct splitting fields. We also show that the situation can be quite different for quadrinomials whose Galois group is not cyclic.

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arXiv2024

Monogenic trinomials and class numbers of related quadratic fields

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N\ge 2$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we investigate the divisibility of the class numbers of quadratic fields ${\mathbb Q}(\sqrt{\delta})$ for certain families of monogenic trinomials $f(x)=x^N+Ax+B$, where $\delta\ne \pm 1$ is a squarefree divisor of the discriminant of $f(x)$.

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arXiv2024

Monogenic Cyclic Cubic Trinomials

A series of recent articles has shown that there exist only three monogenic cyclic quartic trinomials in ${\mathbb Z}[x]$, and they are all of the form $x^4+bx^2+d$. In this article, we conduct an analogous investigation for cubic trinomials in ${\mathbb Z}[x]$. Two irreducible cyclic cubic trinomials are said to be equivalent if their splitting fields are equal. We show that there exist two infinite families of non-equivalent monogenic cyclic cubic trinomials of the form $x^3+Ax+B$. We also show that there exist exactly four monogenic cyclic cubic trinomials of the form $x^3+Ax^2+B$, all of which are equivalent to $x^3-3x+1$.

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arXiv2024

Monogenic cyclic trinomials of the form $x^4+cx+d$

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $n$ that is irreducible over ${\mathbb Q}$ is called cyclic if the Galois group over ${\mathbb Q}$ of $f(x)$ is the cyclic group of order $n$, while $f(x)$ is called monogenic if $\{1,\theta,\theta^2,\ldots, \theta^{n-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this article, we show that there do not exist any monogenic cyclic trinomials of the form $f(x)=x^4+cx+d$. This result, combined with previous work, proves that the only monogenic cyclic quartic trinomials are $x^4-4x^2+2$, $x^4+4x^2+2$ and $x^4-5x^2+5$.

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arXiv2024

The monogenicity and Galois groups of certain reciprocal quintinomials

We say that a monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for ${\mathbb Z}_K$, the ring of integers of $K={\mathbb Q}(\theta)$, where $f(\theta)=0$. For $n\ge 2$, we define the reciprocal quintinomial \[{\mathcal F}_{n,A,B}(x):=x^{2^n}+Ax^{3\cdot 2^{n-2}}+Bx^{2^{n-1}}+Ax^{2^{n-2}}+1\in {\mathbb Z}[x].\] In this article, we extend our previous work on the monogenicity of ${\mathcal F}_{n,A,B}(x)$ to treat the specific previously-unaddressed situation of $A\equiv B\equiv 1\pmod{4}$. Moreover, we determine the Galois group over ${\mathbb Q}$ of ${\mathcal F}_{n,A,B}(x)$ in special cases.

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arXiv2024

Monogenic trinomials of the form $x^4+ax^3+d$ and their Galois groups

Let $f(x)=x^4+ax^3+d\in {\mathbb Z}[x]$, where $ad\ne 0$. Let $C_n$ denote the cyclic group of order $n$, $D_4$ the dihedral group of order 8, and $A_4$ the alternating group of order 12. Assuming that $f(x)$ is monogenic, we give necessary and sufficient conditions involving only $a$ and $d$ to determine the Galois group $G$ of $f(x)$ over ${\mathbb Q}$. In particular, we show that $G=D_4$ if and only if $(a,d)=(\pm 2,2)$, and that $G\not \in \{C_4,C_2\times C_2\}$. Furthermore, we prove that $f(x)$ is monogenic with $G=A_4$ if and only if $a=4k$ and $d=27k^4+1$, where $k\ne 0$ is an integer such that $27k^4+1$ is squarefree. This article extends previous work of the authors on the monogenicity of quartic polynomials and their Galois groups.

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arXiv2024

Monogenic Cyclic Quartic Trinomials

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In this brief note, we prove that there exist exactly three distinct monogenic trinomials of the form $x^4+bx^2+d$ whose Galois group is the cyclic group of order 4.

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arXiv2024

Monogenic Even Octic Polynomials and Their Galois Groups

A monic polynomial $f(x)\in {\mathbb Z}[x]$ of degree $N$ is called monogenic if $f(x)$ is irreducible over ${\mathbb Q}$ and $\{1,\theta,\theta^2,\ldots ,\theta^{N-1}\}$ is a basis for the ring of integers of ${\mathbb Q}(\theta)$, where $f(\theta)=0$. In a series of recent articles, complete classifications of the Galois groups were given for irreducible polynomials \[{\mathcal F}(x):=x^8+ax^4+b\in {\mathbb Z}[x]\] and \[{\mathcal G}(x):=x^8+ax^6+bx^4+ax^2+1\in {\mathbb Z}[x], \quad a\ne 0.\] In this article, for each Galois group $G$ arising in these classifications, we either construct an infinite family of monogenic octic polynomials ${\mathcal F}(x)$ or ${\mathcal G}(x)$ having Galois group $G$, or we prove that at most a finite such family exists. In the finite family situations, we determine all such polynomials. Here, a ``family" means that no two polynomials in the family generate isomorphic octic fields.

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